
\magnification = \magstep1
\input amssym.def            % small letters for UNIX,  not: AMSsym.def
\input epsf.def% \input epsf %for UNIX
\input OurPlainGraphicsMacros
%\input epsf          %\input epsf.def for MAC f"ur BILDER!!

\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum

{\bf Notes} \LF
cross reference =: (C-R)  should be included on occasion\lf
Version March 2001     \LF
Riemann still has a hole.  \lf
 (C-R)  from Tubes to surfaces  \lf
 (C-R)  from Spiral to complex exp  \lf
ODE - ATO: More narrative text is needed.  \lf
conformal maps: Narrative text and C-R needed  \lf
elliptic functions: shorter text needed, C-R to ellipses  \lf




\vskip1.5cm %%%%%%%%%%%%% Text Nr 1

{\bf Pinkall's flat tori in $\Bbb S^3$}

\lf
The tori which we usually see are, from the point of view
of complex analysis, rectangular tori. This means: They have
an orientation reversing symmetry and the set of fixed points of this
symmetry has two components. The well known tori of revolution have
isometric reflections with {\bf two} circles as fixed point sets. Of
course one tries to deform these tori to obtain nonrectangular ones.
Obviously one can destroy the mirror symmetry, but this does not imply
that one gets tori with a nonrectangular complex structure. The first
proof, by Garcia, that one can embed all tori in $\Bbb R^3$ was
nonconstructive and difficult. Pinkall's construction gives completely
explicit flat tori in $\Bbb S^3$. They have a one parameter family of
nonintersecting great circles on them which are parallel in the flat
geometry of the torus. Stereographic projection from  $\Bbb S^3$ to
 $\Bbb R^3$ gives conformal images of these flat tori.
The tori together with the stereographic images of
these great circles are obtained as follows:

\LF(continue with the present text)


\vskip1.5cm %%%%%%%%%%%%% Text Nr 2

{\bf Boy's surface (Bryant-Kusner)}

\lf
See M\"obius Strip first. The non-orientable surfaces are
{\it onesided}, and this can best be understood, if one starts
from a M\"obius Strip. The Klein Bottle is easier to visualize
than the Boy surface. Each meridian of the Boy surface is the
center line of a narrow M\"obius band, for example change in
``Set u,v ranges'': umin = - 0.998,  vmin = 6.1.
\lf
The ``equator'' of the Boy surface is a different M\"obius band,
it has {\it three} half twists instead of one. The standard morph
begins with this M\"obius band and widens it until the Boy
surface is complete:  \lf
$ aa = 0.5,\ vmin = 0, vmax = 2\pi, umax = 1,\
      0.9 \ge umin \ge 0.002$
\lf {\bf Dick: Could you make this the Default? I find ures = 8,
vres = 24 better than the present values 15, 22})
\LF
Boy's surface is really a family of surfaces. Boy, in his dissertation
under Hilbert, constructed this surface
as the first known {\it immersion} of the projective plane.
Being nonorientable implies that no embedding is possible. Boy's surface
has besides its selfintersection curves only one more serious singularity,
namely a triple point. Boy's construction was topological.
\LF
Ap\'ery found algebraically embedded ``Boy's surfaces''. They carry
one-parameter families of ellipses.
\lf
The Bryant-Kusner Boy's surfaces are obtained by an inversion from
a minimal surface in $\Bbb R^3$. The minimal surface is an immersion
of $\Bbb S^2 - \{6\ points\}$ such that antipodal points have the same
image in $\Bbb R^3$. The six punctures are three antipodal pairs, and
the minimal surface has so called {\it planar ends} at these punctures.
In this context it is important that the puncture, which the inversion
of a planar end has, can be closed {\it smoothly} by adding one point.
The closing of the three pairs of antipodal ends thus gives a tripple
point on the surface obtained by inversion.
\lf
The formulas are as follows:

\LF(continue with the present text)


\vskip1.5cm %%%%%%%%%%%%% Text Nr 3

{\bf Pseudosphere}

\lf
The Pseudosphere is a surface of revolution of Gaussian curvature -1,
or in other words, the product of its principal curvatures is -1. On
a surface of revolution this translates into a simple analytic
property: Parametrize the meridian curve by arc length
$s \mapsto \big(r(s),h(s)\big),\ r'^2 + h'^2 = 1$. Then $r()$
is a solution of the differential equation $r'' = r$. Consquently
$h$ is also known, $h = \int^s \sqrt{1 - r'(u)^2}du$.
\LF
The Pseudosphere is best known because its intrinsic geometry is
hyperbolic, the meridians are a family of asymptotic geodesics
and the orthogonal lattitudes are therefore a geodesically parallel
family of ``horocycles'', i.e. $\!$limits of circles as their midpoints
converge to the limit point of the asymptotic geodesics.
\LF
This Pseudosphere is obtained by the construction which relates
solutions of the Sine-Gordon equation to surfaces of Gaussian
curvature -1, here the solution:

\centerline{$q(u,v) := 4\cdot \arctan(\exp(u))$.} \noindent
The obtained parametrization has another remarkable property: The
diagonal curves in {\bf all} the parameter quadrilaterals have the
same length! Nets used for fishing also have such equiquadrilaterals
as meshes; the mathematical term is ``Tchebycheff net''. Such
Tchebycheff nets exist on all surfaces which are isometric immersions
of (portions of) the hyperbolic plane. This fact plays a key role in the
proof of Hilbert's theorem which says: There is no smooth isometric
immersion of the whole hyperbolic plane into euclidean threespace.
\LF
See also the ``Tractrix'' under {\it Planar Curves}.


\vskip1.5cm %%%%%%%%%%%%% Text Nr 4 see Nr 22 - 24

{\bf Ellipse}

\lf
(formula as in present text)

\lf
The generalized cycloids are particularly interesting for circles
(i.e., $aa = bb$). Recall that negative values of the rolling radius
$hh$ gives curves on the outside, positive radii role on the inside.
The {\bf Cardiod} is obtained with $aa = bb = -hh$ if you choose
{\it Show generalized cycloids} from the ``Action Menu''.
Similarly, the {\bf Astroid} is obtained with $+hh = 0.5*aa$. The
coordinate axes cut segments of constant length
from the tangents of the astroid.
\LF(Pictures illustrating the construction of the ellipse are
missing. Note that metric properties carry over to ellipses on the
sphere.)


\vskip1.5cm %%%%%%%%%%%%% Text Nr 5

{\bf Userdefined by Curvature}

\lf
(No entry yet)

\lf
A planar curve (parametrized by arc length) can be constructed
from its curvature function $s\mapsto\kappa(s)$ as follows:
First take the antiderivative
$\alpha(s) := \int^s \kappa(\sigma)d\sigma$. Then choose an
initial point $p$, an initial tangent $\dot c(0)$ vector and
an orthonormal basis $ e_1 = \dot c(0),\ e_2$. The definition of
curvature (namely $ \kappa := |\ddot c|$, plus a sign convention)
implies
 $$ \dot c(s) = e_1\cdot \cos\alpha(s) +  e_2\cdot \sin\alpha(s).$$
One more integration $c(s) = p + \int_0^s \dot c(\sigma)d\sigma$
determines the curve.
\lf
This description explains why the curvature is also called the
``rotation speed'' of the tangent vector field $\dot c(s)$.


\vskip1.5cm %%%%%%%%%%%%% Text Nr 6

{\bf M\"obius Strip}

\lf
The M\"obius Strip is the most famous {\it onesided} surface. Try
from the view-menu ``distinguish sides by color'', you will see
that the sides are not distinguished -- because there is only one,
follow the band around.
\lf
M\"obius Strips can be found on all non-orientable surfaces. To
see it on  the  Klein Bottle select from the settings-menu
``Set t,u,v ranges'' and put  umin = - 0.4,  umax = + 0.4  .
On the BOY surface there are even two different kinds. To see
one with {\it three} halftwists change to  umin = 0.9; this is a
band with the ``equator'' of the BOY surface as its center line.
Bands with meridians as center curves are ordinary M\"obius bands,
change the u,v-ranges to umin = -0.998,  vmin = 6.1 : This is best
seen with ``distinguish sides by color'' from the view-menu.
\lf
(On the Steiner surface and on the Crosscap one can also find
M\"obius Strips. These however are
not embedded and  therefore not so recognizable. In Math jargon:
These surfaces are not immersed and the singularity is on the
M\"obius band.)

\vskip1.5cm %%%%%%%%%%%%% Text Nr 7

{\bf Klein Bottle}

\lf
See M\"obius Strip first. The non-orientable surfaces are
{\it onesided}, and this can best be understood, if one starts
from a M\"obius Strip.
\lf
Imagine that we modify a torus by rotating a figure 8 instead
of a circle. If we color the two sides of this surface differently
then one loop of the figure 8 has one color, the other loop the
other color. (One can make such a surface in the surface menu, by
taking the Lemniscate as the meridian curve in the entry
``User Defined (Rotation)''.
{\bf Dick: Could you make this the Default?})
\lf
The present Klein Bottle is obtained with one further modification:
Rotate the meridian figure 8 in its plane by 180 degrees while
the plane is rotated. One can see the figure 8 better if one
changes in ``Set u,v ranges''  vmin = 0.5. Use ``Distinguish
Sides by Color'' in this cut open view. One can see this in a morph
with \lf
$ aa= 3,\ umin=0,\ umax=2*\pi,\ vmin=0,\  0.5 \le umax \le 2*\pi$.
\lf
The default morph starts from the M\"obius Strip  \lf
 $ -0.4 \le u \le 0.4,\ 0 \le v \le 2*\pi,  aa = 3 $  \lf
and increases the width of the M\"obius
Strip until it closes to the Klein Bottle at  \lf
  $ - \pi \le u \le \pi.$
{\bf (Dick: Please change the Default}) \lf
Actually, there are three different kinds of Klein Bottles which
cannot be deformed into each other: (i) The present one, where the
figure 8 rotates to the left in its plane; (ii) The mirror image
of the present one, where the figure 8 rotates clockwise; (iii)
a Klein Bottle with mirror symmetry, glas models show this case.

\vskip1.5cm %%%%%%%%%%%%% Text Nr 8

{\bf Tractrix}

\lf
The Tractrix is a curve with the following nice interpretation:
If a dog owner starts at the origin walking along the y-axis with
his dog standing initially on the x-axis at the distance aa away from
the owner, with the leash being tight. The definition of the
Tractrix is that the dog follows the owner unwillingly, i.e.
keeping the leash tight. This means mathematically, that the
leash is always tangent to the path of the dog, i.e. the length
of the tangent segment from the Tractrix to the y-axis has
{\bf constant} length aa. \lf
The Pascal code for the Tractrix is:  \lf
P.x := aa*sin(t)   \lf
P.y := aa*(cos(t) + ln(tan(t/2)) )
\LF
The Tractrix has a famous surface of revolution, called the
Pseudosphere: Rotation around the y-axis gives a surface with
Gaussian curvature -1. This means that the Pseudosphere can
be considered as a portion of the Hyperbolic Plane, which
was discovered in the 19th century: It satisfies all the axioms
of Euclidean Geometry except the axiom of Parallels.



\vskip1.5cm %%%%%%%%%%%%% Text Nr 9

{\bf Rolling Circles on Circles}

\lf
Many of the curves, which have individual names, were already
considered (and named) by the Greeks. A large class of these
can be obtained by rolling one circle on the inside or the outside
of another circle. The Greeks were interested in rolling
constructions, because this was their main tool to describe the
motions of the planets (Ptolemai). The following curves from
this program can be obtained by rolling constructions:
\lf
{\bf Cycloid, Ellipse, Asteroid, Cardioid, Limacon.}
\lf
Not all geometric properties of these curves follow easily
from their definition as rolling curve, but in some cases
the connection with complex functions (Conformal Category)
does.
\LF
{\bf Cycloids} are generated by rolling a circle on a straight
line. The Pascal code for such a cycloid is  \lf
P.x := aa*t - bb*sin(t)          \lf
P.y := aa - bb*sin(t) ,   aa = bb \lf
Cycloids have other cycloids of the same size as evolute (Action
Menu: ``Show Osculating Circles with Normals'').  This fact is
responsible for Huyghen's cycloid pendulum to have a period
independent of the amplitude of the oscillation.
\LF
{\bf Ellipses} are obtained if {\it inside} a circle of radius aa
another circle of radius  r = hh = 0.5*aa rolls and then traces
a curve with a radial stick of length  R = ii*r:
\lf
The Pascal code for such an ellipse is  \lf
P.x := (R+r)*cos(t)          \lf
P.y := (R-r)*sin(t)          \lf
In the visualization of the complex map  $z \to z + 1/z$ in
Polar Coordinates the image of the circle of Radius R is such
an ellipse with $r = 1/R$.
\LF
{\bf Asteroids} are obtained if {\it inside} a circle of radius aa
another circle of radius  r = hh = 0.25*aa rolls and then traces
a curve with a radial stick of length  R = ii*r = r:
\lf
The Pascal code for such an Asteroid is   \lf
P.x := (aa-r)*cos(t) + R*cos(4*t)         \lf
P.y := (aa-r)*sin(t) - R*sin(4*t)         \lf
Asteroids can also be obtained by rolling the {\it larger} circle
of radius r = hh = 0.75*aa (put gg = 0 in this case).
Another geometric construction of the Asteroids uses the fact
that the length of the segment of each tangent between the
x-axis and the y-axis has {\bf constant} length.
\LF
{\bf Cardioids and Limacons} are obtained if
{\it outside} a circle of radius aa
another circle of radius  r = hh = - aa rolls and then traces
a curve with a radial stick of length  R = ii*r, ii = 1 for
the Cardioids, ii $>$ for the Limacons:
\lf
The Pascal code for Cardioids and Limacons is   \lf
P.x := (aa+r)*cos(t) + R*cos(2*t)         \lf
P.y := (aa+r)*sin(t) + R*sin(2*t)         \lf
The Cardioids and Limacons can also be obtained by rolling the
larger circle of radius r = hh = + 2*aa; now ii $<$ 1 for the
Limacons. Note that the fixed circle is {\it inside}
the larger rolling circle. \lf
The evolute of the Cardioid
(Action Menu: ``Show Osculating Circles with Normals'')
is a smaller Cardioid. The image of the unit circle
unter the complex map $z \to w = (z^2 + 2z)$ is a Cardioid; images
of larger circles are Limacons. Inverses $z \to 1/w(z) $ of Limacons
are figure 8 shaped, one of them is a Lemniscate.
\LF
{\bf Nephroids} are generated by rolling a circle of one radius
outside of a second circle of twice the radius, as the program
demonstrates. With $R = 3r$ we thus have the Parametrization as\lf
Pascal Code for Nephroids \lf
P.x := R*cos(t) + r*cos(3*t)         \lf
P.y := R*sin(t) + r*sin(3*t)         \lf
As with Cardioids and Limacons one can also make the radius for the
drawing stick shorter or longer: Choose in the Menu ''Circle'' and
set Parameters $aa = 1, hh = -0.5, ii = 1$ for the Nephroid and
$ii > 1$ for its looping relatives.
\lf
The complex map $ z \to z^3 + 3z$ maps the unit circle to such a
Nephriod. To see this, in the Conformal Map Category, select
$z \to z^{ee} + ee\cdot z$ from the Conformal Map Menu, then choose
Set Parameters from the Settings Menu and put $ee = 3$.
\lf
Back in the Plane Curves Category, select Nephroid and then in the
Action Menu: ``Show Osculating Circles with Normals''. The Normals
envelope a smaller Nephroid.


\vskip1.5cm %%%%%%%%%%%%% Text Nr 10

{\bf Astroid}

\lf
The Pascal code of a parametrization of the Astroid is \lf
$ P.x := aa*\cos(t)^3   \lf
  P.y := aa*\sin(t)^3 $.
\lf
Therefore this Astroid can also be described by the equation  \lf
  $ |x|^{2/3}  +  |y|^{2/3}  =  aa^{2/3}$. \lf
A nice geometric property of the Astroid is that its tangents, when extended
until they cut the x-axis and the y-axis, all have the same length. This means,
if one leans a ladder under different angles  against a wall, then the envelope
of the ladder's positions is part of an Astroid. -- This shows that the
diagonal of the Astroid is twice as long as its waist-diameter. Moreover,
the normals of an Astroid envelope an Astroid twice its size, thus giving
a ruler construction: Intersect each ladder (between the x-axis and  the
y-axis) for the smaller Astroid with the orthogonal and twice as long ladder
(between the 45-degree lines) for the larger Astroid.
\lf
If one selects an Ellipse and sets parameters so that $ x=aa/bb = 1.4656$
is the solution of $x^2(x-1) = 1$ and chooses the Action ``Show osculating
circles with normals'', then these normals envelope an Astroid in the same
way as the ladder.


\vskip1.5cm %%%%%%%%%%%%% Text Nr 11

{\bf Complex Map $ z \to z^{aa}$}

\lf
In school we mainly learn about one-dimensional functions
$x \to f(x)$ and we visualize them in the plane by drawing
their graphs $\{(x,f(x)); x \hbox{ from the domain of $f$ in } \Bbb R \}$.
Since the complex numbers are already visualized by the
{\bf Gaussian Plane} it is impossible to see the graphs of complex
functions in our three-dimensional world, and we need other means for
their visualization.
\lf
This program employs the same idea which already Maxwell used
extensively: A grid in the domain (or preimage) of the function is mapped
to give an (usually curved) image grid in the range. Comparing the
two (colored) grids allows to develop a good intuition how the function
$f$ maps the points in the domain to points in the range.
\lf
The most important fact about complex differentiable maps is that they
preserve angles, i.e. the angles between intersecting curves in
the domain are the same as the angles between the image curves. This
can best be visualized if the domain grid consists of little squares.
The range grid consists then of curves which intersect each other
orthogonally. This fact alone is not enough for the map $f$ to preserve
all angles. But the diagonals of a square also intersect orthogonally,
therefore the diagonal curves in the grids are also orthogonally
intersecting curves. Allthough the diagonal curves are not shown, their
orthogonality has an important visible consequence: The grid meshes
look like slightly curved squares and they look more like squares,
if one maps smaller grid meshes. This fact allows to ``see'' whether
a map is angle preserving (= conformal); look at the function
$ z \to conj(z) +aa\cdot z^2$, it is not conformal and the grid meshes
are so obviously far from almost square that one cannot miss to see this.
\lf
As with one-dimensional functions it is a fact that those points where
the derivative vanishes, dominate the overall appearance of the
image grid very much. Most obviously, near points $a$ with $f'(a)=0$
the gridmeshes get very small and as a consequence, the grid lines
usually are strongly curved. If one looks more closely then one notices
that the angle between curves through $a$ is NOT the same as the
angle between the image curves through $f(a)$ (recall: $f'(a)=0$).
One should first look at the behaviour of the simple quadratic
function $z\to z^2$ near $a=0$, both in Cartesian and in Polar
coordinates. One sees that a rectangle, which touches $a=0$ from
one side is folded around $0$, and one also sees in Polar coordinates
that the angle between rays from $0$ gets {\bf doubled}.
\lf
Fortunately, the behaviour of this simple function is already very
typical. We only need to exclude the case where also $f''(a)=0$ and then
we can say: In a small neighbourhood of a point $a$ where
$f'(a)=0,f''(a)\ne 0$ the mapping behaviour is almost the same as
that of the prototype function $z\to z^2$ near $a=0$. In fact, the closer
one looks the smaller the visible differences get! To get aquainted
with this way of looking at the figures study the functions
$z\to z+1/z$ and $z\to z^2+2z$, in particular enlarge them near the
critical point (press SHIFT and move the mouse).


\vskip1.5cm %%%%%%%%%%%%% Text Nr 12  see also Nr. 27

{\bf Parabola}

\lf
The Pascal code of the Parabola is \lf
$ P.x := t^2/4p    \lf
  P.y := t $,                 \lf
where $p = aa/4.$
\lf
The Parobola therefore visualizes the graphs of the two functions
$ y(x) := \sqrt{4p\cdot x}$ and $ x(y) := y^2/4p$.
\lf
The vertical line $x=-p$ is called the directrix and the point
$(x,y)=(p,0)$ is called focal point of the Parabola. The distance from a
point $(x,y= \sqrt{4p\cdot x})$ on the Parabola to the directrix is $(x+p)$,
and this is the same distance as from $(x,y= \sqrt{4p\cdot x})$
to the focal point $(p,0)$, because $(x-p)^2 + y^2 = (x+p)^2$.
\lf
The point $(p,0)$ is called ''focal point'', because light rays which
come in parallel to the x-axis are reflected off the Parabola so that
they continue to the focal point. This fact is illustrated in the program.
It gives the following ruler construction of the Parabola:
\lf
Prepare the construction by drawing x-axis, y-axis, directrix and focal
point F.
Then draw any line parallel to the x-axis and intersect it with the directrix
in a point   S. The line orthogonal to the connection SF and through its
midpoint
is the tangent of the Parabola and intersects therefore the incoming ray
in the point of the Parabola which we wanted to find.
\lf
{\bf Dick:} The same construction works for Ellipse and Hyperbola, if the
directrix is replaced by a circle of radius 2*a around one focal point. The
curve is the set of points which have the same distance from this circle
and the
other focal point. I wait whether you like the Parabola text before I write
the same for the other two.
\lf
{\bf Dick:} If you make the normals *longer* than only to the evolute
and if you deselect Parametrization by arclength, than you see a fascinating
hexagonal intersection pattern: If y1 + y2 + y3 = 0, then the normals
intersect!



\vskip1.5cm %%%%%%%%%%%%% Text Nr 13

{\bf Complex Map $ z \to z^2$}

\lf
Look at the general descriptions in ''About this Category'' for what
to see, what to expect and what to do.
\lf
As with one-dimensional functions it is a fact that those points where
the derivative vanishes, dominate the overall appearance of the
image grid very much. Most obviously, near points $a$ with $f'(a)=0$
the gridmeshes get very small and as a consequence, the grid lines
usually are strongly curved. If one looks more closely then one notices
that the angle between curves through $a$ is NOT the same as the
angle between the image curves through $f(a)$ (recall: $f'(a)=0$).
We will find this general description applicable to many examples.
\lf
One should first look at the behaviour of the simple quadratic
function $z\to z^2$ near $a=0$, both in Cartesian and in Polar
coordinates. One sees that a rectangle, which touches $a=0$ from
one side is folded around $0$ with strongly curved parameter lines,
and one also sees in Polar coordinates
that the angle between rays from $0$ gets {\bf doubled}. The image
grid in the Cartesian case consists of two families of orthogonal
intersecting parabolas. \lf
One should return to this prototype picture after one has seen others
like $z\to z+1/z$, $z\to z^2 + 2z$ and even the Elliptic functions
and looked at the behaviour near their critical points ($f'(a)=0$).
\lf
The first examples to look at, {\bf both in Cartesian and Polar Grids}
are
\lf
$\phantom{.} \hskip1.5in z\to z^2,\ z\to 1/z,\ z\to \sqrt{z},\ z\to e^z$.


\vskip1.5cm %%%%%%%%%%%%% Text Nr 14

{\bf Complex Map $ z \to 1/z$}

\lf
Look at the function $z\to z^2$ and its ATO first.
\lf
The function $ z \to 1/z$ should be looked at  both in Cartesian
and Polar Grids.
Notice first: Real axis, imaginary axis and unit circle are mapped
into themselves. The upper half plane goes to the lower half plane,
the inside of the unit circle to its outside, and vice versa. This
is maybe best seen in the (default) Conformal Polar Grid.
\lf{\bf Dick:} Can you fatten the unit circle? \lf
Or if not then at least  make u-range $0.25 < u < 4$? \lf
In the Cartesian Grid one should in particular observe that all
straight parameter lines (in the domain) are mapped to circles
(some exceptions, like the real axis, remain lines). The behaviour
of these circles near zero can be looked at as an image of the
behaviour of the standard Cartesian Grid near infinity. In fact {\bf
all} circles are mapped to circles or lines.
\lf
Examples to look at after this are
\lf
$\phantom{.} \hskip1.1in z\to (az+b)/(cz+d),\ z\to (z+cc)/(1+\bar{cc}z)$.
\lf
Both can be obtained from $z\to 1/z$ by composition with translations
$z\to z+a$ or scaled rotations $z\to a\cdot z$. Therefore all of these
``M\"obius transformations'' map circles and lines to circles and lines. 


\vskip1.5cm %%%%%%%%%%%%% Text Nr 15

{\bf Complex Map $ z\to \sqrt{z}$}

\lf
Look at the functions $z\to z^2$, $ z\to 1/z$ and their ATOs first.
\lf
The function $ z\to \sqrt{z}$ should be looked at  both in Cartesian
and Polar Grids. \lf
Notice: Since this function is the inverse of $z\to z^2$ one should
see the same things: Circles around $0$ go to circles around $0$,
radial lines from $0$ go to radial lines from $0$, but now with
{\bf half} the angle between them (since we look at the inverse map).
A neigbourhood of $0$ was very much contracted by $z\to z^2$, now we
see the opposite, the distance of points from zero is increased very
much (beyond any Lipschitz bound). \lf
A more complicated aspect is the fact, that $z\to \sqrt{z}$ is not a
well defined map, since all $z\ne 0$ have two different square roots
$\pm\sqrt{z}$.
The Pascal function $\sqrt{z}$ maps the upper half plane to the first
quadrant, the (strict) lower halfplane to the fourth quadrant, the
negative real axis to the positive imaginary axis --  so there is no
continuity from above to below the negative real axis (which is
therefore called a ''branch cut'').
\lf
The Cartesian grid lines are mapped to two families of {\bf hyperbolae}
which intersect each other orthogonally.



\vskip1.5cm %%%%%%%%%%%%% Text Nr 16

{\bf Complex Map $ z\to e^z$, the complex Exponential}

\lf
Look at the functions $z\to z^2$, $ z\to 1/z$ and their ATOs first.
\lf
{\bf Dick:} change v-range to $-3.1 < v < 3.1$ and vRes from 20 to 30
\lf
The complex exponential function $z\to e^z$ is one of the most
marvellous functions around. It shares with the real function
$x \to \exp(x)$ the differential equation $\exp' = \exp$ and the
functional equation $\exp(z+w)=\exp(z)\cdot\exp(w)$. This implies that
one can understand the complex Exponential in terms of real functions:
Putting $z = x + i\cdot y$ we have \lf
$\phantom{.} \hskip0.4in
\exp(x+i\cdot y) = \exp(x)\cdot\exp(i\cdot y) =
\exp(x)\cdot\cos(y) + i\cdot\exp(x)\cdot\sin(y)$.
\lf
This says that the standard Cartesian Grid is mapped ''conformally''
(= angle preserving) to a Polar Grid, the parallels to the real axis
are mapped to the radial lines; and segments of length $2\pi$ which are
parallel to the imaginary axis are mapped to circles around $0$. This
function is therefore used to make, in the Action Menu, the
Conformal Polar Grid. Observe how justified it is to describe the image
grid as ''made out of curved small squares''.
\lf
If you have seen $z\to e^z$ and $z\to z+1/z$ then look at $z\to \sin(z)$.


\vskip1.5cm %%%%%%%%%%%%% Text Nr 17

{\bf Complex Map $ z\to (a\cdot z + b)/(c\cdot z +d)$ }

\lf
Look at the functions $z\to z^2$, $ z\to 1/z$ and their ATOs first.
\lf
{\bf Dick:} change to \ \  aa = 1 and bb = -1
\lf
These functions are called {\bf M\"obius transformations}, also ``fractional
linear maps''. They differ from $z\to 1/z$ by composition with translations
$z\to z+a$ or scaled rotations $z\to a\cdot z$. As discussed for $z\to 1/z$
they map lines and circles to lines and circles.
\lf
The default special case is $ z\to (z-1)/(z+1)$. It is best understood
in the (default) Conformal Polar Grid. Since it maps $0$ to $-1$ and
$\infty$ to $+1$, one can see the Polar coordinate centers moved from
$0,\infty$ to $-1,+1$. This picture is the beginning of understanding
the complex plane (''Gaussian Plane'') plus infinity also as the
''Riemann Sphere''.
\lf
See the other M\"obius transformation from the selection Menu.



\vskip1.5cm %%%%%%%%%%%%% Text Nr 18

{\bf Complex Map $ z\to z^{ee} + ee\cdot z$, default $z\to z^2+2z$ }

\lf
Look at the functions $z\to z^2$, $ z\to 1/z$ and their ATOs first.
\lf
Of course, since $z^2 + 2z +1 = (z+1)^2$, this function is not very
different from the first example $z\to z^2$. But the change puts the
critical point to $-1$ on the unit circle ($f'(-1)=0$). Therefore, if
one looks what this map does to a Polar Grid, one can study the behaviour
near the critical point $z=-1$ with a different grid picture than in
the first example. Circles outside the unit circle are mapped to
Limacons (Plane Curves Category) which wind around $-1$ twice. The
unit circle is mapped to a Cardioid and one can see the interior
angle of 180 degrees of the unit circle at $-1$ mapped to the interior
angle of 360 degrees of the Cardioid at $-1$. Also one can see that a
neigbourhood of $-1$ is strongly contracted by this function.
 \lf
See the function $z\to z+1/z$ next.



\vskip1.5cm %%%%%%%%%%%%% Text Nr 19

{\bf Complex Map $ z\to z + 1/z$ }

\lf
Look at the functions $z\to z^2$, $ z\to 1/z$,\ $z\to z^2+2z$,\
$ z\to e^z$ and their ATOs first.
\lf
This function is best applied to a Conformal Polar Grid. The image
of the outside of the unit circle is the same as the image of the
inside of the unit circle, namely the full plane minus the segment
$[-2,2]$. The unit circle is mapped to this intervall, each interior
point $w = 2\cdot\cos(\phi)\in[-2,2]$ appears twice as image point,
namely of $z = \exp(\pm i\phi)$. \lf
The default choice shows how the outside of the unit disk is mapped
to the outside of the interval $[-2,2]$. If we note that $f'(\pm 1)=0$
then we understand this behaviour: The interior 180 degree angle at these
critical points $\pm 1$ of the outside domain is again {\bf doubled}
to become the angle of the image domain (outside $[-2,2]$) at $\pm 2$.
\lf
The domain circles are mapped to the image Ellipses:  \lf
$\phantom{.} \hskip0.4in z(\phi) = R\cdot\exp(i\phi)  \mapsto
 w(\phi) = (R+1/R)\cdot\cos(\phi) + i\cdot(R-1/R)\cdot\sin(\phi)$,
\lf
and the domain radii are mapped to the image Hyperbolae:  \lf
$\phantom{.} \hskip0.4in z(R) = R\cdot\exp(i\phi)  \mapsto
 w(R) = (R+1/R)\cdot\cos(\phi) + i\cdot(R-1/R)\cdot\sin(\phi)$.
\lf
The image grid therefore consists of a family of Ellipses which
intersect a family of Hyperbolae orthogonally, and all these
Conic Sections (see Plane Curves Category) have the
{\bf same Focal Points}, namely at $\pm 2$.

\vskip1.5cm %%%%%%%%%%%%% Text Nr 20

{\bf Complex Map $ z\to \sin(z)$ }

\lf
Look at the functions $z\to z^2$, $ z\to 1/z$,\ $z\to z^2+2z$,\
$ z\to e^z$ and their ATOs first.
\lf
While for the onedimensional real functions $x\to \exp(x),
x\to \sin(x)$ the behaviour is far apart (exp is convex and positive,
sin is periodic and bounded), as complex functions they are very
closely related:
$$ \sin(z) = (\exp(iz) - \exp(-iz))/2i.$$
This explains, why the image grid under sin of  the default Cartesian
grid looks exactly the same as the image grid under $z\to z+1/z$
applied to a Conformal Polar Grid outside the unit circle:
\lf
Put $w(z) :=\exp(iz)/i$, then $\sin(z) = (w(z)+1/w(z)))/2$. And also
recall that exp maps the standard Cartesian Grid to the Conformal
Polar Grid around $0$. The parameter curves in the image grid of sin
are therefore the same orthogonal and confocal ellipses and hyperbola
as in the image of $z\to z+1/z$.

\vskip1.5cm %%%%%%%%%%%%% Text Nr 21

{\bf Lemniscate }

\lf
The Lemniscate is a figure 8 curve with the implicit equation
$(x^2 + y^2)^2 = x^2 - y^2$. Divide this by $r^2:= x^2 + y^2$
to get the polar form $ r^2 = \cos(\phi)^2 - \sin(\phi)^2$.
Parametrizations are not unique, here is one:
\lf
$P.x :=\hskip1cm \cos(t)/(1+\sin(t)^2)  \lf
 P.y :=\sin(t)\ \cos(t)/(1+\sin(t)^2)$. \lf
The points $F_1,F_2:=\pm\sqrt{2}$ are called Focal points of the
Lemniscate because of the special property:\hskip1cm
$|P-F_1|\cdot|P-F_2| = |F_1-F_2|^2/4$.
\LF
If one takes the complex square root of a circle which touches
the $y$-axis from the right at $0$ then one also obtains a Lemniscate.
(Choose in the Conformal Category: $z \to \sqrt{z}$ and then in the
Action Menu: Choose Circle by Mouse, touching the $y$-axis at $0$.)
\LF
Often the inversion map: $ (x,y)\mapsto (x,y)/(x^2+y^2)$  makes
out of some interesting curve another one. Applying this idea
to the parametrization of the Lemniscate gives the curve: \lf
$x =1/\cos(t),\ y=\sin(t)/\cos(t)$, which is a hyperbola, since
$x^2-y^2=1$. So we could have obtained the Lemniscate by inversion
from the standard hyperbola.
\LF
We note that not every figure 8 curve is a Lemniscate, another
figure 8 is obtained by the simpler parametrization:
\lf
$x(t) := \cos(t) \lf
 y(t) := \sin(t)\cdot \cos(t);$ \lf
it has the implicit equation $y^2 = x^2(1-x^2)$.



\vskip1.5cm %%%%%%%%%%%%% Text Nr 22

{\bf Ellipse}

\lf
See also Parabola, Hyperbola, Conic Sections and their ATOs.
\lf
The most common parametric equations for an Ellipse with semi-axes aa
and bb are:

\lf
$x(t) = aa \cos(t)  \lf
 y(t) = bb \sin(t) ,\hskip1cm  0\le t \le 2\pi$.

\lf
(Here, $aa$ and $bb$ are assumed positive. The larger is called the 
semi-major axis length and the smaller the semi-minor axis length.)

The corresponding implicit equation is  \lf
$ (x/aa)^2 + (y/bb)^2 = 1$.

\lf
A geometric definition of the Ellipse, which has been used to
shape flower beds in parks, is:  \lf
An Ellipse is the set of points for which the {\bf sum of the distances}
from two focal points is a constant $L$ equal to twice the semi-major 
axis length.   \lf
Gardeners used
to connect the two focal points by a cord of length $L$, pulled
the cord tight with a stick which then drew the boundary of the flower
bed. Another version of this definition is: An Ellipse is the set of
points which have {\bf equal distance} from a circle of radius $L$
and a (focal) point inside the circle. Both these definitions are illustrated
in the program. \lf
Tangent and normal lines to the Ellipse are the angle bisectors of the two focal
lines.  This also says that rays coming out of one focal point are reflected
off the Ellipse towards the other focal point. Therefore one can build
elliptically shaped 'whispering halls', where very quiet words which
are spoken at one focal point can be heard only close to the other
focal point.\lf
To add a simple proof we show that the tangent leaves the ellipse on one
side; more precisely, we show that for every other point on the tangent the
sum of the distances to the two focal points $F_1, F_2$
is more than the length $L$ of the major axis. (In the display: $F=F_2$.)
 Pick any point $Q$ on the tangent, join it to the two focal points
and reflect the segment $QF$ in the tangent, giving another segment $QS$.
Now $F_1QS$ is only a radial straight segment if $Q$ is the point of tangency,
else $F_1QS$ is by the triangle inequality longer than the radius $F_1S$
(of length $L$) of the circle around $F_1$.
\LF
An Ellipse can also be obtained by a rolling construction: Inside a circle of
radius $aa$ another circle of radius r  = hh = 0.5*aa rolls and traces
the Ellipse with a stick of radius R = ii*r. The parametric equation resulting
from this construction is:

\lf
$ x(t) = (R + r)*\cos(t) \lf
  y(t) = (R - r)*\sin(t) $

\lf
This is related to the visualization of the complex map $z\to z+1/z$ in Polar
Coordinates, the image of the circle of radius $R$ is such an ellipse with
$r=1/R$.
\lf
Such rolling constructions are reached with the Menu entry 'Circle' and then
the Action Menu 'Draw Generalized Cycloids'. Recall that negative values of the
rolling radius $hh$ gives curves on the outside, positive radii ($hh < aa$)
on the inside of the fixed circle. \lf
Other rolling curves are:  \lf
Cycloid, Astroid, Cardioid, Limacon, Nephroid.


\vskip1.5cm %%%%%%%%%%%%% Text Nr 24

{\bf Conic Sections, 2D construction}

\lf
See also Parabola, Ellipse, Hyperbola and their ATOs.
\LF
A cone of revolution, for example $\{(x,y,z); x^2 + y^2 = m\cdot z^2\}$,
is one of the simplest surfaces. Its intersections with planes are called
conic sections. Apart from pairs of lines these conic sections are
Parabolae, Ellipses or Hyperbolae. These curves have also other geometric
definitions (like: The set of points which have the same distance from a focal
point and a circle), see their Menu entries.
\lf
On the other hand, they are also more robust than these definitions show:
Photographic images of conic sections are again conic sections; or in a
completely different formulation: The intersections of planes and \lf
\phantom{.}\hskip0.5cm ``quadratic cones'', e.g.
$\{(x,y,z); a\cdot x^2 + b\cdot y^2 + c\cdot z^2 +d\cdot xy+e\cdot yz = 0\}$,
\lf
are {\bf not} more complicated than planar sections of circular cones but
are the same old Parabolae, Ellipses or Hyperbolae as above. A special case
of this robustness is the fact that orthogonal projections of conic sections
in 3-space are again conic sections. This is illustrated in the program as
follows:
\LF
Interprete the illustration as if it showed level lines on a hiking map.
The equidistant parallel lines are the level lines of a sloping plane;
the smaller the distance between these level lines the steeper the plane.
The equidistant concentric circles are the level lines of a circular cone,
as for example an antlion  would dig in sandy ground; without height numbers
written next to the level lines we can of course not decide whether the
circular level lines represent a conical mountain or a conical hole in
the ground. We suggest that the blue level line and the vertex of the cone
are at height zero and the other levels are higher up so that the cone is a
hole. \lf
The intersection curve between plane and cone has then an easy pointwise
construction: Simply intersect level lines of the same height on the two
surfaces. (These are lines with the same color in the program illustration.)
This construction reveals a new geometric property of the intersection curve
on the map, of this conic section:
\par {\narrower\noindent
Take the ratio of the distances from a point on the curve, (i) to the level
line at height 0 of the plane (called {\it directrix})
and (ii) to the vertex at height zero of the cone (called {\it focus}).
{\it This ratio is the same} as the ratio of adjacent level lines of plane
and cone and therefore the same {\it for all points of this conic section}.
\par}



\vskip1.5cm %%%%%%%%%%%%% Text Nr 25

{\bf Conic Sections, Kepler orbits}

\lf
See also Parabola, Ellipse, Hyperbola and their ATOs.
\LF
For many properties of the conic sections a parametrization is not relevant.
However, when Kepler discovered that planets and comets travel on conic sections
around the sun then this discovery came with a companion: the speed on the orbit 
is such  that angular momentum is preserved. In more elementary terms: the radial
connection from the sun to the planet sweeps out equal areas in equal times. With
the 3dfs demo we explain geometrically how this celestial parametrization is
connected with the focal properties of conic sections. Here we give the {\it algebraic
explanation} first.
\LF
An ellipse, parametrized as {\it affine image of a circle} and translated to the left is
$$ E(\varphi) := (a\cos\varphi - e,\ b\sin \varphi).$$
If we choose $e:=\sqrt{a^2-b^2}$ then we have $|E(\varphi)|=(a-e\cos\varphi)$. This
gives the connection with the oldest definition of an ellipse: The sum of the distances
from $E(\varphi)$ to the two points $(\pm e, 0)$ is $2a$. 
\lf
Next, the derivative $E'$,
twice the area, denoted $A$, swept out by the position vector and the derivative  of  $A$
are
$$ E'(\varphi) = (-a\sin\varphi, b\cos\varphi),\ 
   A(\varphi) =\int_0^\varphi \det(E(\varphi),E'(\varphi))d\varphi,\
   A'(\varphi) = b(a- e\cos\varphi).$$
We denote the inverse function of $A(\varphi)$ by $\Phi(A)$, i.e., 
$$\Phi(A(\varphi)) = \varphi,\hskip1cm   \Phi'(A)= {1\over b(a-e\cos\Phi)}.$$ 
By Kepler's  second law $A$ is proportional to time and we can write down the
Kepler parametrization of the ellipse and its velocity:
$$ K(A) := E(\Phi(A)),\hskip1cm   K'(A) = E'(\Phi(A))\cdot \Phi'(A).$$
We need one line of computation to simplify the kinetic energy:
$$\eqalign{
     {1\over 2}K'(A)^2 &= 
      {a^2\sin^2\varphi + b^2\sin^2\varphi \over 2b^2(a-e\cos\varphi)^2}
      = {a^2 -e^2\cos^2\varphi \over 2b^2(a-e\cos\varphi)^2} 
    = {(a+e\cos\varphi) \over 2b^2(a-e\cos\varphi)} 
\cr
    &= {a\over b^2}\cdot {1\over a-e\cos\varphi} - {1\over 2b^2}.
     = {a\over b^2}\cdot {1\over |E(A)|} - {1\over 2b^2}.
\cr}$$
This shows that, in our units (we took twice the swept out area as time), the potential
energy (up to a constant) can be read off by using {\it kinetic + potential enrgy =
const}: 
$$ \hbox{Potential energy at orbit point }E(A)\hskip5mm \hbox{ equals }\hskip5mm  
    -{a\over b^2}\cdot {1\over |E(A)|},$$
which is the famous $1/r$ law.
\LF
Now we present a {\it geometric proof}. The starting point is the determination of
the correct orbital speed  by the property that the product of the speed $|v|$ 
with the distance $p$ of the tangent line from the center is the constant angular
momentum, Kepler's second law. Of course we
can illustrate such a fact only if we also represent the size of velocities by the 
length of segments and we have to keep in mind that segments which illustrate a
length and segments which illustrate a velocity are interpretated with different units.
\lf
Recall the following theorem about circles: if two secants of a circle intersect 
then the product of the subsegments of one secant ist the same as the product of
the subsegments of the other secant.
\lf
This will be applied to the circle the radius of which is the length $2a$ of the
major axis. (The midpoint is the other focus, not the sun.) The two secants 
intersect in the focus representing the sun: one secant is an extension of the major 
axis the other is perpendicular to the tangent line. The subsegments of the first secant
have the lengths $2a-2e$ and $2a+2e$, where $2e$ is the distance between the foci.
The subsegments of the second secant have one length $2p$ and one labeled $|v|$. The
circle theorem says: $(2a-2e)\cdot(2a+2e)= 2p\cdot|v|$. 
Since the left side is constant we can interprete
the segment labeled $|v|$ as representing the {\it correct orbital speed}.
\LF
Now that we know at each point of the orbit the correct speed we can deduce Newton's
$1/r$-law for the gravitational potential, if we use {\it kinetic energy plus
potential energy equals constant total energ}. In the illustration we have two
similar right triangles, the small one has hypothenuse $=r$ and one other side $=p$,
the big one has as hypothenuse a circle diameter of length $4a$ and the corresponding
other side has length $2p+|v|$. Now we use the above 
$const:=(2a-2e)\cdot(2a+2e)= 2p\cdot|v|$
to eliminate $p$ from the proportion:
$$  p:r = (2p+|v|) : 4a $$
This gives 
$$ 2a/r = 1 + |v|/2p = 1 + v^2/const.$$
Up to physical constants (units) $v^2$ is the kinetic energy, so that (again up to
units) $-1/r$ is the potential energy -- since such a potential makes kinetic plus
potential energy constant.

\hbox{     \vbox{
           \hbox{\hskip1cm\epsfysize=2.6in {\epsfbox{KeplerEllipse.eps}} }
  \hbox{\hskip0.5cm A Kepler Ellipse with construction of proper speed and potential. } 
   % \hbox{\hskip0.5cm Other text.}
                }
      }


\vskip1.5cm \goodbreak %%%%%%%%%%%%% Text Nr 26

{\bf Conic Sections and the Dandelin Spheres}

\lf
See also Category Planar Curves: Parabola, Ellipse, Hyperbola, Conic Section and their
ATOs.
\LF
(1) Parabolae, Ellipses and Hyperbolae have a three-dimensional definition as planar
sections of (usually circular) cones, and they have a two-dimensional definition as loci
having equal distance from a point and a line resp. as loci having the sum (or the
difference) from two points being constant. The Dandelin Spheres explain why the different
definitions give the same curves.
The program illustration shows:
\lf
(i) A cone and two inscribed spheres.  The two spheres touch the cone in 
    highlighted circles $C_1,C_2$. \lf
(ii) A plane which intersects the cone in a highlighted curve $E$ and which also
tangentially intersects the two spheres in highlighted points $F_1,F_2$.
\lf
The illustration is to explain that for each point $P$ on the intersection curve $E$ the
sum of its distances to $F_1,F_2$ is constant, namely equal to the distance between the
circles $C_1,C_2$ on the cone. For this note that all tangent segments from $P$ to each of
the spheres have the {\bf same} length, so that the distances from $P$ to $F_j$ and from
$P$ to $C_j,\ j=1,2 $ are equal. This shows that $E$ is an ellipse: 

\centerline{$ |P-F_1|+|P-F_2| = \hbox{distance}(C_1,C_2). $}

\lf
(2) In the Planar Curve ATO on Conic Sections we have introduced the directrix of a conic
section as the line such that the {\bf ratio} of the distances from a point $P$ on the
conic  to the Focus $F_j$ and to the directrix $D_j,\ j=1,2$ is constant ($=1$ for
Parabolae, $<1$ for Ellipses and $>1$ for Hyperbolae). In the program illustration these
directrices are highlighted, they are the two intersection lines of the plane of $E$
with the two parallel planes of the circles $C_1,C_2$. We said already that the
distances from $P$ to $F_j$ and from $P$ to $C_j,\ j=1,2 $ are equal for all $P$. We
therefore have to prove that the ratios 

\centerline{$ \hbox{distance}(P,D_j) : \hbox{distance}(P,C_j). $}

\noindent
are constant, namely equal to the ratio 
$ \hbox{distance}(D_1,D_2) : \hbox{distance}(C_1,C_2). $
But this follows since the two triangles made out of the segments 
$\overline{PD_j}$ and $\overline{PC_j}$ (and closed by a segment of $D_j$) for $j=1$ and
$j=2$ are similar.



\vskip1.5cm %%%%%%%%%%%%% Text Nr 27  earlier version Nr 12

{\bf Parabola}

\lf
See also: Ellipse, Hyperbola, Conic Section and their ATOs. \lf
And in the Category Surfaces see: Conic Sections and Dandelin Spheres
\LF
The Pascal code of the Parabola is \lf
$ P.x := t^2/4p    \lf
  P.y := t $,                 \lf
where $p = aa/4.$
\lf
The Parobola therefore visualizes the graphs of the two functions
$ y(x) := \sqrt{4p\cdot x}$ and $ x(y) := y^2/4p$.
\lf
The vertical line $x=-p$ is called the directrix and the point
$(x,y)=(p,0)$ is called focal point of the Parabola. The distance from a
point $(x,y= \sqrt{4p\cdot x})$ on the Parabola to the directrix is $(x+p)$,
and this is the same distance as from $(x,y= \sqrt{4p\cdot x})$
to the focal point $(p,0)$, because $(x-p)^2 + y^2 = (x+p)^2$.
\lf
The point $(p,0)$ is called ''focal point'', because light rays which
come in parallel to the x-axis are reflected off the Parabola so that
they continue to the focal point. This fact is illustrated in the program.
It gives the following ruler construction of the Parabola:
\lf
Prepare the construction by drawing x-axis, y-axis, directrix and focal
point F.
Then draw any line parallel to the x-axis and intersect it with the directrix
in a point   S. The line orthogonal to the connection SF and through its
midpoint
is the tangent of the Parabola and intersects therefore the incoming ray
in the point of the Parabola which we wanted to find.
\lf
The same construction works for Ellipse and Hyperbola, if the
directrix is replaced by a circle of radius 2*a around one focal point. The
curve is the set of points which have the same distance from this circle
and the other focal point. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Text Nr 12 ends here %%%%%%%%%%%%%%%%
\LF
The Action Menu of the Parabola has the entry ``Show Normals Through Mouse Point''.
This action illustrates an unexpected property of the Parabola. One may already
be surprised that at the intersection points of normals always {\bf three} normals
meet. We know no other curve which is accompanied by such a net of normals. The
surprise should increase if one looks at the $y$-coordinates of the parabola
points from where three such intersecting normals originate: these $y$-coordinates
add up to $0$! In other words, the intersection behaviour of the normals reflects
the addition on the $y$-axis. \lf
The normals of the Parabola are tangents to its evolute. This evolute is a singular
cubic curve, see: Cuspidal Cubic. The intersection property of the parabola normals 
is a limiting  case of the less simple addition which exists for any cubic curve.



\vskip1.5cm %%%%%%%%%%%%% Text Nr 28

{\bf Complex Map $ z\to \log z$, the complex Logarithm}

\lf
Look at the function $z\to e^z$ and its ATO first.
\lf
The complex Logarithm tries to be the inverse function of the complex
Exponential. But since $\exp$ is $2\pi i$-periodic, such an inverse can
only be a multivalued function. From the differential equation $\exp' = \exp$ 
follows that the derivative of the inverse is not multivalued and in fact
very simple: 

\centerline{$ \log'(z) = 1/z$.}

\noindent
Integration of the geometric series 

\centerline{
$1/z = 1/(1-(1-z)) = \sum_k (1-z)^k = \left(\,\sum_k-(1-z)^{k+1}/(k+1)\right)'$}

\noindent
gives the Taylor expansion around $1$ of $\log$. The so called ``principal value''
of the complex Logarithm is defined in the whole plane, but slit along the negative
real axis, for example by integrating the derivative $ \log'(z) = 1/z$
in that simply connected domain along any path which starts at $1$.
\lf
Different values of $\log z$ differ by integer multiples of $2\pi i$, e.g.
$i = \exp(\pi i/2)$ implies $\log i = \pi i/2 + 2\pi i\cdot\Bbb Z$.


\vskip1.5cm %%%%%%%%%%%%% Text Nr 29

{\bf Complex Map $ z\to (z + cc)/(1+\overline{cc}\cdot z)$, M\"obius transformation
of the unit disk.}

\lf
Look at the M\"obius transformation $z\to (a\cdot z + b)/(c\cdot z +d)$ and its ATO
first.
\lf
This function maps the interior of the unit disk bijectively to itsself, for every
choice of $cc$ with $|cc|<1$. The behaviour outside of the unit disk is obtained by
reflection  in the unit circle, i.e., $z\to 1/\bar z$.
\lf
These maps have another interesting interpretation: they are isometries for the
``hyperbolic metric'' on the unit disk. To understand this further, imagine that
the unit disk is a map of this twodimensional hyperbolic world and that the scale
of this map is not a constant but equals $1/(1-z\bar z)$. This means that we do not
obtain the length of a curve $t\to z(t)$ as in the Euclidean plane by the
integral $\int |z'(t)|dt$, but we have to take the scale into account and have
the hyperbolic length as $\int |z'(t)|/(1-|z(t)|^2)dt$. It is this 
hyperbolic length of curves which is left invariant by the ``hyperbolic
translations'' $ z\to (z + cc)/(1+\overline{cc}\cdot z)$.
\lf
Locally the Pseudosphere (Category: Surfaces) has the same hyperbolic geometry.
\vfill
\vskip1.5cm %%%%%%%%%%%%% Text Nr 30
\eject 
{\bf Addition on Cubic Curves.}

\lf
See also the Action Menu of the Parabola ``Show Normals through Mouse Point''
and the comments in the ATO.
\lf
As an introductory example view the unit circle as a group. Then the addition
of angles $\phi\in (\Bbb R$ mod $ 2\pi)$ gets translated via the parametrization
$x = \cos(\phi), y=\sin(\phi)$ into 

\centerline{$(x_1,y_1)\oplus(x_2,y_2):= 
(x_1x_2 - y_1y_2,\ x_1y_2 + x_2y_1)$.}             

\noindent
Once this addition law is known one does not need the transcendental functions
$\sin$ and $\cos$ to ``add'' points on the circle. Even to do this addition with
ruler and compass is easy. And it is amusing to note that the Pythagorean (or
rational) points of the circle are a subgroup, e.g. 
$(3/5, 4/5)\oplus(12/13,5/13) = ( (36-20)/65, (15+48)/65)$.
\LF
In a similar way there exists a geometric addition on cubic curves, and if the
cubic is parametrized with appropriate functions (defined either on $\Bbb C$,
or on $\Bbb C/2\pi\Bbb Z$, or on $\Bbb C/\Gamma,\ \Gamma$ a lattice in $\Bbb C$)
then the well known addition in the domain is, under the special parametrization,
the same as the geometric addition on the cubic.
\LF
The simplest instance is when the cubic is the graph of a cubic polynomial without 
quadratic term: $y = x^3+mz+c$. Then, if we have two points $(x_1,y_1),(x_2,y_2)$  
on this cubic and join them by a line, this line intersects the graph in a third
point $(x_3,y_3)$ such that $x_1+x_2+x_3=0$. This gives a geometric definition of
addition on this cubic graph.

\hbox{     \vbox{\vskip-0cm
           \hbox{\hskip1cm\epsfysize=3.7in {\epsfbox{CubicGraphP.eps}} }
 \centerline{ Addition on a polynomial cubic graph without quadratic term. } 
   \centerline{Every line intersects so that $x_1 + x_2 +x_3 = 0$.
                Note discrete subgroup.}
                }
      }

\lf
Similarly, let us map $\Bbb C$ bijectively onto the Cuspidal Cubic by
$z\mapsto (z^2,z^3)$. In this case, 
if we have $z_1+z_2+z_3=0$, then the tangents at the three
points $(z_j^2,z_j^3)$ are concurrent---we have seen this as a property
of the Parabola, because the Cuspidal Cubic is the evolute of the Parabola. One
can also see the previous colinearity as reflecting addition, because the three
points $(z_j^2,z_j^3), j=1,2,3,$ of this cubic lie on a line if 
$1/z_1 +1/z_2 +1/z_3 =0$.

\hbox{     \vbox{
           \hbox{\hskip1cm\epsfysize=3.8in {\epsfbox{CubicCusp.eps}} }    
  \centerline{Addition on the cuspidal cubic $z\mapsto (z^2,z^3)$.
        Note the discrete subgroup. } 
  \centerline{If $z_1+z_2+z_3=0$, then the tangents at these three points 
              are concurrent.}
 \centerline{If $1/z_1+1/z_2+1/z_3=0$, then these three points lie on a straigt
line.}
                }
      }

\noindent
The next case is the group $\Bbb C/2\pi\Bbb Z$. The trigonometric functions
identify points in $\Bbb C$ mod $2\pi $. We map this group to a cubic curve by
$x:=\tan(z/2),\ y:=\sin(z)$, so that $y = 2x/(x^2+1)$ and this cubic is again a
graph. The addition theorems $\tan(z+w) =(\tan(z) +\tan(w))/(1-\tan(z)\tan(w))$
and $\sin(z+w) = \sin(z)\cos(w)+\cos(z)\sin(w)$ with 
$\cos(z) = 1 - 2\sin(z/2)^2 = 1 - \sin(z)\cdot\tan(z/2)$ again give an addition
on this cubic graph: it is a geometric addition because the three points
$(x_j,y_j)$ lie on one line iff $z_1+z_2+z_3=0$. The name ``geometric addition'' is
even more justified because the third point $(x_3,y_3)$ can be constructed with
ruler and compass from the other two. In fact, for repeated additions a ruler
suffices: As a preparation we have to add to all points in sight the 2-division
point $(\infty, 0)=(\tan(\pi/2),\sin(\pi))$ as follows: 
$(x,y)\oplus(\infty, 0) = (-1/x,-y)$. One needs ruler and unit circle for this.
But then the lines through $(x_1,y_1),(x_2,y_2)$ and 
$(x_1,y_1)\oplus(\infty, 0), (x_2,y_2)\oplus(\infty, 0)$ intersect in the
point $(x_3,y_3)=-(x_1,y_1)\oplus(x_2,y_2)$.
\lf

\hbox{     \vbox{
           \hbox{\hskip0cm\epsfysize=4.4in {\epsfbox{CubicGraphR.eps}} }
   \centerline{Addition group $\Bbb S^1$ on a cubic that is also the graph
    of $x\mapsto y = 2x/(x^2+1)$, } 
   \centerline{parametrized by $x:=\tan(z/2),\ y:=\sin(z) $.
Note the finite discrete subgroup.}
   \centerline{The point $(\infty,0)= (\tan(\pi/2), \sin(\pi))$ 
           at infinity is the only point of order 2.}
                }
      }
\lf
So far we have seen the circle part of the cylinder group $\Bbb C/2\pi\Bbb Z$.
To see a generator of the cylinder we replace $t,x,y$ by $it,ix,iy$, then we obtain 
$x:=\tanh(z/2),\ y:=\sinh(z)$, so that $y = 2x/(1-x^2)$. The component of the graph
through $0$ is a subgroup isomorphic to $\Bbb R$. It represents one generator of
the cylinder. The other two components represent the opposite generator with one
point missing: the 2-division point opposite $0$ is the point $(\infty,0)$ on this
cubic. This allows the same ruler construction of addition as before, except for a
sign change in $(x,y)\oplus(\infty, 0) = (+1/x,-y)$ (because $1/i=-i$).
\lf

\hbox{     \vbox{
           \hbox{\hskip0.1cm\epsfysize=4.0in {\epsfbox{CubicGraphiR.eps}} }
   \centerline{Addition group $\Bbb R\cup\Bbb R$ on a cubic that is also
               the graph of $x\mapsto y = 2x/(1-x^2)$ and is}  
   \centerline{parametrized by $x:=\tanh(z/2),\ y:=\sinh(z)$. $(\infty,0)$
               is the only point of finite order.}
   \centerline{Note the infinite discrete subgroup with one finite subgroup
               of order 2.}
                }
      }

\lf
Finally we come to the group $\Bbb C/\Gamma$. The parametrizing functions of the
previous example, $\tan(z/2)$ and $\sin(z)$, have to be replaced by 
$\Gamma$-invariant, ``doubly periodic'' functions (also called elliptic 
functions), and the simplest of these are those
of degree two, as maps from the torus $T^2:=\Bbb C/\Gamma$ to the Riemann
sphere $\Bbb S^2 = \Bbb C\cup\{\infty\}$. Two facts are important: 
\item{(i)} 
  Pairs of such functions satisfy cubic equations such as
  $(w^2+1)v = \hbox{const}\cdot(v^2-1)w$.
  The solution set of any cubic equation is called a cubic curve.
\item{(ii)} 
  There are addition formulas, analogous to those for $\sin$ and $\cos$,
  which determine the pair  
  $(v(z_1+z_2), w(z_1+z_2))$ from the pair $(v(z_1), w(z_1))$ and the pair
  $(v(z_2), w(z_2))$.

\noindent
It turns out that these addition formulas are again ``geometric'' as in the
previous cases, namely, the three pairs   
 $(v(z_1), w(z_1)),\ (v(z_2), w(z_2)),\ (-v(z_1+z_2), -w(z_1+z_2))$
lie on a line. Therefore we can again define addition on the cubic geometrically:
\item{}
{\it Join the points to be added by a line and take the third point of intersection
with the cubic as the negative of the sum}.

\noindent
The addition formulas are simple enough so that the geometric addition is again
a ``ruler and compass construction''. The compass is only needed to add
2-division points as in the previous case, all further additions can be done by
only intersecting lines.

\hbox{     \vbox{
           \hbox{\hskip0.1cm\epsfysize=4in {\epsfbox{CubicGeneral.eps}} }
  \centerline{The addition on the general cubic $w+1/w =2(v-1/v)$
              is given by the formula: }

      $$ (w_1,v_1)\ominus (w_2,v_2) =
                ({w_1+w_2 \over 1-w_1w_2 }\cdot{v_1-v_2 \over v_1+v_2 },
                 {  1+w_1w_2 \over  1-w_1w_2 }\cdot{v_1-v_2 \over  1-v_1v_2}).
$$  
               }
      }
     \centerline{Notice the discrete subgroup.\hskip7.7cm}
\LF
The elliptic functions $v,w$, that parametrize the above cubic curve, have many
properties that could be taken as their definition. For example they are 
numerically accessible since they are solutions of the following pair of
differential equations:
$$\eqalign{
{v'\over v} = w'(0)\left({1\over w} - w\right),        \hskip1.2cm
{w'\over w} = v'(0)\left({1\over v} + v\right),
\cr}$$
with $v'(0)/w'(0) = -2$ for the above cubic. These imply equations for each
function so that some similarity with the trigonometric case, namely
$(\sin')^2 = 1 - \sin^2$, becomes apparent:
$$\leqalignno{
\left({v'\over v}\right)^2 = w'(0)^2\left({1\over w} - w\right)^2 &=   
  w'(0)^2\left(({1\over w} + w)^2 - 4\right) =  
  v'(0)^2\left(({1\over v} - v)^2 \right)  - 4w'(0)^2  
\cr
 (v')^2 &= v'(0)^2\left((1 - v^2)^2 -4{w'(0)^2 \over v'(0)^2}\cdot v^2\right).
  &\hbox{hence}
\cr}$$

\bye
